Stabilizer of Subgroup Action is Identity

Theorem

Let $\struct {G, \circ}$ be a group whose identity is $e$.

Let $\struct {H, \circ}$ be a subgroup of $G$.

Let $*: H \times G \to G$ be the subgroup action defined for all $h \in H, g \in G$ as:

$\forall h \in H, g \in G: h * g := h \circ g$

The stabilizer of $x \in G$ is $\set e$:

$\Stab x = \set e$

From Subgroup Action is Group Action we have that $*$ is a group action.

Let $x \in G$.

Then:

$\ds \Stab x$

$=$

$\ds \set {h \in H: h * x = x}$

Definition of Stabilizer

$\ds $

$=$

$\ds \set {h \in H: h \circ x = x}$

Definition of $*$

$\ds $

$=$

$\ds \set {h \in H: h = x \circ x^{-1} }$

$\ds $

$=$

$\ds \set {h \in H: h = e}$

$\ds $

$=$

$\ds \set e$

Hence the result, by definition of right coset.

$\blacksquare$

1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 5.6$. Stabilizers: Example $111$
1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Sylow Theorems: $\S 54 \alpha$